Extremal properties of linear dynamic systems controlled by Dirac’s impulse

• Autorzy: Białas Stanisław , Górecki Henryk , Zaczyk Mieczysław

Identyfikatory

DOI

10.34768/amcs-2020-0006

• Języki publikacji: EN

Abstrakty

EN

The paper concerns the properties of linear dynamical systems described by linear differential equations, excited by the Dirac delta function. A differential equation of the form a_n x⁽ⁿ⁾ (t) + ∙∙∙ a₁ x’(t) + a₀ x(t) = b_m u<sup(m)</sup> (t) + ∙∙∙ + b₁ u’(t) + b₀ u(t) is considered with a_i, b_j >0. In the paper we assume that the polynomials M_n(s) = a_nsⁿ + ∙∙∙ + a₁s + a₀ and L_m(s) = b_ms^m + ∙∙∙ + b₁s + b₀ partly interlace. The solution of the above equation is denoted by x(t, L_m, M_n). It is proved that the function x(t, L_m, M_n) is nonnegative for t ∊ (0, ∞) , and does not have more than one local extremum in the interval (0, ∞) (Theorems 1, 3 and 4). Besides, certain relationships are proved which occur between local extrema of the function x(t, L_m, M_n), depending on the degree of the polynomial M_n(s) or L_m(s) (Theorems 5 and 6).

Słowa kluczowe

EN

extremal properties; Dirac's impulse; linear system; transfer function

PL

właściwości ekstremalne; impuls Diraca; układ liniowy; funkcja przenoszenia

• Wydawca: Oficyna Wydawnicza Uniwersytetu Zielonogórskiego
• Czasopismo: International Journal of Applied Mathematics and Computer Science
• Rocznik: 2020
• Tom: Vol. 30, no. 1
• Strony: 75--81
• Opis fizyczny: Bibliogr. 5 poz., wykr.

Twórcy

autor

Białas Stanisław

• School of Banking and Management, ul. Armii Krajowej 4, 30-150 Cracow, Poland

autor

Górecki Henryk

• Department of Automatics and Robotics, AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Cracow, Poland

autor

Zaczyk Mieczysław

• Department of Automatics and Robotics, AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Cracow, Poland

Bibliografia

• [1] Górecki, H. (2018). Optimization and Control of Dynamic Systems, Springer, Cham.
• [2] Górecki, H. and Zaczyk, M. (2013). Design of systems with extremal dynamic properties, Bulletin of the Polish Academy of Sciences: Technical Sciences 61(3): 563–567.
• [3] Kaczorek, T. (2002). Positive 1D and 2D Systems, Springer, London.
• [4] Kaczorek, T. (2018). A new method for determination of positive realizations of linear continuous-time systems, Bulletin of the Polish Academy of Sciences: Technical Sciences 66(5): 605–611.
• [5] Osiowski, J. (1965). An Outline of Operator Calculus. Theory and Applications in Electrical Engineering, WNT, Warsaw, (in Polish).

Uwagi

PL

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2020).